Seminars and Colloquia by Series

Artin fans in tropical geometry

Series
Algebra Seminar
Time
Monday, September 22, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Martin UlirschBrown University
Recent work by J. and N. Giansiracusa, myself, and O. Lorscheid suggests that the tropical geometry of a toric variety $X$, or more generally of a logarithmic scheme $X$, can be formalized as a "Berkovich analytification" of a scheme over the field $\mathbb{F}_1$ with one element that is canonically associated to $X$.The goal of this talk is to introduce the theory of Artin fans, originally due to D. Abramovich and J. Wise, which can be used to lift rather unwieldy $\mathbb{F}_1$-geometric objects to the more familiar realm of algebraic stacks. Artin fans are \'etale locally isomorphic to quotient stacks of toric varieties by their big tori and their glueing data has a completely combinatorial description in terms of Kato fans.I am going to explain how to use the ideas surrounding the notion of Artin fans to study tropicalization maps associated to toric varieties and logarithmic schemes. Surprisingly these techniques allow us to give a reinterpretation of Tevelev's theory of tropical compactifications that can be generalized to compactifications of subvarieties in logarithmically smooth compactifcations of smooth varieties. For example, we can introduce definitions of tropical pairs and schoen varieties in terms of Artin fans that are equivalent to Tevelev's notions.

Determinantal representations of hyperbolic curves

Series
Algebra Seminar
Time
Friday, September 19, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Daniel PlaumannUniversität Konstanz
We study symmetric determinantal representations of real hyperbolic curves in the projective plane. Such representations always exist by the Helton-Vinnikov theorem but are hard to compute in practice. In this talk, we will discuss some of the underlying algebraic geometry and show how to use polynomial homotopy continuation to find numerical solutions. (Joint work with Anton Leykin).

Groebner bases for fields with valuations

Series
Algebra Seminar
Time
Monday, June 30, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Anders JensenAarhus University
In this talk we discuss a recent paper by Andrew Chan and Diane Maclagan on Groebner bases for fields, where the valuation of the coefficients is taken into account, when defining initial terms. For these orderings the usual division algorithm does not terminate, and ideas from standard bases needs to be introduced. Groebner bases for fields with valuations play an important role in tropical geometry, where they can be used to compute tropical varieties of a larger class of polynomial ideals than usual Groebner bases.

The Tate-Shafarevich group of the Legendre curve

Series
Algebra Seminar
Time
Monday, May 5, 2014 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Professor Doug UlmerGeorgia Tech
We study the Legendre elliptic curve E: y^2=x(x+1)(x+t) over the field F_p(t) and its extensions K_d=F_p(mu_d*t^(1/d)). When d has the form p^f+1, in previous work we exhibited explicit points on E which generate a group V of large rank and finite index in the full Mordell-Weil group E(K_d), and we showed that the square of the index is the order of the Tate-Shafarevich group; moreover, the index is a power of p. In this talk we will explain how to use p-adic cohomology to compute the Tate-Shafarevich group and the quotient E(K_d)/V as modules over an appropriate group ring.

Bounded gaps between primes in Chebotarev sets

Series
Algebra Seminar
Time
Monday, April 28, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jesse ThornerEmory University
A new and exciting breakthrough due to Maynard establishes that there exist infinitely many pairs of distinct primes $p_1,p_2$ with $|p_1-p_2|\leq 600$ as a consequence of the Bombieri-Vinogradov Theorem. We apply his general method to the setting of Chebotarev sets of primes. We study applications of these bounded gaps with an emphasis on ranks of prime quadratic twists of elliptic curves over $\mathbb{Q}$, congruence properties of the Fourier coefficients of normalized Hecke eigenforms, and representations of primes by binary quadratic forms.

Torus actions and faithful tropicalisations

Series
Algebra Seminar
Time
Friday, April 11, 2014 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jan DraismaTU Eindhoven
Given a closed subvariety X of affine space A^n, there is a surjective map from the analytification of X to its tropicalisation. The natural question arises, whether this map has a continuous section. Recent work by Baker, Payne, and Rabinoff treats the case of curves, and even more recent work by Cueto, Haebich, and Werner treats Grassmannians of 2-spaces. I will sketch how one can often construct such sections when X is obtained from a linear space smeared around by a coordinate torus action. In particular, this gives a new, more geometric proof for the Grassmannian of 2-spaces; and it also applies to some determinantal varieties. (Joint work with Elisa Postinghel)

High Rank Quadratic Twists of Elliptic Curves

Series
Algebra Seminar
Time
Monday, March 24, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Nick RogersDepartment of Defense
A notorious open problem in arithmetic geometry asks whether ranks ofelliptic curves are unbounded in families of quadratic twists. A proof ineither direction seems well beyond the reach of current techniques, butcomputation can provide evidence one way or the other. In this talk wedescribe two approaches for searching for high rank twists: the squarefreesieve, due to Gouvea and Mazur, and recursion on the prime factorization ofthe twist parameter, which uses 2-descents to trim the search tree. Recentadvances in techniques for Selmer group computations have enabled analysisof a much larger search region; a large computation combining these ideas,conducted by Mark Watkins, has uncovered many new rank 7 twists of$X_0(32): y^2 = x^3 - x$, but no rank 8 examples. We'll also describe aheuristic argument due to Andrew Granville that an elliptic curve hasfinitely many (and typically zero) quadratic twists of rank at least 8.

The essential skeleton of a degeneration of algebraic varieties

Series
Algebra Seminar
Time
Wednesday, March 12, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Johannes NicaiseKU Leuven
I will explain the construction of the essential skeleton of a one-parameter degeneration of algebraic varieties, which is a simplicial space encoding the geometry of the degeneration, and I will prove that it coincides with the skeleton of a good minimal dlt-model of the degeneration if the relative canonical sheaf is semi-ample. These results, contained in joint work with Mircea Mustata and Chenyang Xu, provide some interesting connections between Berkovich geometry and the Minimal Model Program.

Singular Learning Theory

Series
Algebra Seminar
Time
Monday, March 10, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Elizabeth GrossNCSU
Bayesian approaches to statistical model selection requires the evaluation of the marginal likelihood integral, which, in general, is difficult to obtain. When the statistical model is regular, it is well-known that the marginal likelihood integral can be approximated using a function of the maximized log-likelihood function and the dimension of the model. When the model is singular, Sumio Watanabe has shown that an approximation of the marginal likelihood integral can be obtained through resolution of singularities, a result that has intimately tied machine learning and Bayesian model selection to computational algebraic geometry. This talk will be an introduction to singular learning theory with the factor analysis model as a running example.

Buildings and Berkovich spaces

Series
Algebra Seminar
Time
Wednesday, March 5, 2014 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 254
Speaker
Annette WernerJohann Wolfgang Goethe-Universität (Frankfurt)
The goal of this talk is to show that Bruhat-Tits buildings can be investigated with analytic geometry. After introducing the theory of Bruhat-Tits buildings we show that they can be embedded in a natural way into Berkovich analytic flag varieties. The image of the building is contained in an open subset which in the case of projective space is Drinfeld's well-known p-adic upper half plane. In this way we can compactify buildings in a natural way.

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